Question

In a certain group of $75$ students, $16$ students are taking physics, geography and english; $24$ students are taking physics and geography, $30$ students are taking physics and english; and $22$ students are taking geography and English. However, $7$ students are taking only physics, $10$ students are taking only geography and $5$ students are taking only english. How many of these students are taking physics and english, but not geography?

Answer 1

Let $A$ represents numbers of students taking physics, $B$, $\to$ students taking Geography

$C\to$ students taking English

Numbers of student $=75$

$n(A\cap B\cap C)$ $=16$

$n(A\cap B)$ $=24$

$n(A\cap C)$ $=30$

$n(B\cap C)$ $=22$

number of students who take physics and mathematics only $=$ number of student who take physics and English$-$ number of students who take a three subjects

$=$ $n(A\cap C)-n(A\cap B\cap C)$

$=30-16$

$=14$